The document discusses risk measurements according to Basel 3.5, specifically evaluating Value at Risk (VaR) and Expected Shortfall (ES). It summarizes that Basel 3.5 transitions from VaR to ES as the standard risk measure. The aim is to see if this represents an improvement in market risk management. Empirical analysis on S&P 500 returns finds ES at 97.5% percentile captures tail risk better than VaR at 99%, confirming Basel's choice. Backtesting also shows ES estimates are more reliable than VaR for determining capital requirements.

1. UNIVERSITÀ DEGLI STUDI DI SALERNO
Dipartimento di Scienze Economiche e
Statistiche
Corso di Laurea Magistrale in Economia e Finanza
Tesi in
Analisi dei mercati finanziari
Risk measurements according to Basel 3.5:
an empirical evaluation of Value at Risk and
Expected Shortfall
Relatore
Ch.ma Prof.ssa
Alessandra Amendola
Candidato
Luca Gravina
Matr. 0222201156

2. Basel 3.5 and FRTB
• The Basel Committee on Banking Supervision (BCBS), is a committee, established in 1974, of the
Bank for International Settlement (BIS). Its task is to regulate the banks’ risk management and setting
the capital standards for financial institutions.
• After the 2008 financial crisis, BCBS chose to revisit fundamental principles for effective risk
management and issued in 2010 a new regulatory framework called “Basel III”.
• Specifically, Basel III introduced fundamental changes in the market risk management, in a specific
framework, the Fundamental Review of the Trading Book (FRTB), become effective in 2012, and then
it underwent several changes until it was completed in January 2016. However, certain elements were
subsequently revised by the Basel Committee in January 2019 in a paper called “Minimum Capital
Requirement for Market Risk”.
• The implementation deadline of the latest version of FRTB standard is set for 1 January 2022.
• The market risk regulatory reform is inserted in a larger framework of reforms for all the risks, called
“Basel 3.5”

3. Transition from VaR to Expected
Shortfall
The latest FRTB provides several instructions that the banks must follow, from the
boundaries between banking and trading book up to revised standardised and internal
approaches to estimate the amount of capital required to cover potential losses in a
future period of stress.
But the main change on which we focus is the transition from VaR to Expected Shortfall
as standard risk measure to compute the capital requirement for the financial
intermediaries allowed to use the internal approach.
The aim of the work is to find out, in general terms, if this choice represents an
improvement in the market risk management or not.

4. Value at Risk

5. Expected Shortfall

6. The data
Close Prices Log-returns
Mean Median Min Max
1597.8 1385.7 676.5 2964.3
Std Dev Coeff. of Var. Skewness Kurtosis
565.1152 0.3536468 0.7919352 2.518554
Mean Median Min Max
0.0003001 0.0007113 -0.1276522 0.1095720
Std Dev Coeff. of Var. Skewness Kurtosis
0.01216755 40.53825 -0.5472382 17.25021
S&P500 index (^GSPC) from 02.01.2003 to 01.07.2019 on a daily base, n. 4152 observations. The sample voluntarily
excludes, in this part, the prices from July 2019 to July 2020, the period of COVID-19 pandemic.

7. Model identification
The model which best fits the data is a MA(1)-GARCH(1,1) with a skewed
Student's t distribution for standardized residuals.
The expression of the model to be considered will therefore be:

8. VaR: Empirical results
Now, we compute and compare the parametric daily Value at Risk on
S&P500 returns using different models and confidence intervals.
Specifically, the analysis compare the VaR at 1%, 2.5%, and 5%, using first
the volatility extracted by a MA(1)-GARCH(1,1) with a skewed Student’s t
distribution, and then using the volatility estimated by the Exponential
Weighted Moving Average (Riskmetrics)
VaR Comparison
Alpha 1% Alpha 2.5% Alpha 5%
MA(1)-GARCH(1,1) -3.35% -2.39% -1.69%
EWMA with normal distribution -2.19%* -1.85%* -1.56%

9. MA(1)-GARCH(1,1) VaR

10. EWMA VaR

11. VaR Backtesting
Backtesting Evaluation MA(1)_GARCH(1,1)
Alpha 1% Alpha 2.5% Alpha 5%
Expected VaR
Exceed
41 103 207
Actual VaR
Exceed
39 116 204
Unconditional Coverage 95% H_0 = Correct
Exceedances
LR.uc Statistic 0.156 1.423 0.064
LR.uc Critical 3.841 3.841 3.841
LR.uc p-value 0.692 0.232 0.799
Reject Null NO NO NO
Conditional Coverage 95% H_0 = Correct and
Independent Exceedances
LR.cc Statistic 3.817 1.598 0.064
LR.cc Critical 5.991 5.991 5.991
LR.cc p-value 0.148 0.449 0.968
Reject Null NO NO NO
Backtesting Evaluation EWMA
Alpha 1% Alpha 2.5% Alpha 5%
Expected VaR
Exceed
41 103 207
Actual VaR
Exceed
87 153 224
Unconditional Coverage 95% H_0 = Correct
Exceedances
LR.uc Statistic 38.280 20.944 1.339
LR.uc Critical 3.841 3.841 3.841
LR.uc p-value 6.126e-10 4.728e-06 0.247
Reject Null YES YES NO
Conditional Coverage 95% H_0 = Correct and
Independent Exceedances
LR.cc Statistic 38.947 21.273 1.340
LR.cc Critical 5.991 5.991 5.991
LR.cc p-value 3.488e-09 2.401e-05 0.511
Reject Null YES YES NO
Once computed the VaR, it must be evaluated, through statistical tools that allow to measure the quality of
different estimates obtained using different approaches, in order to compare them with each other.
The statistical tests able to do that are defined as backtesting, because the evaluations are feasible only ex-
post, once we have access to historical values

12. MA(1)-GARCH(1,1) VaR Backtesting

13. EWMA VaR Backtesting

14. 1 Year VaR and Backtesting
The supervisor’s control at bank-wide level encompasses a range of
possible responses, classified into three 1% prior year VaR backtesting
zones, distinguished by colours into a hierarchy of riskiness.
Backtesting Zones
Number of exceptions Backtesting dependent
multiplier
Green
0
1
2
3
4
1.50
1.50
1.50
1.50
1.50
Amber
5
6
7
8
9
1.70
1.76
1.83
1.88
1.92
Red 10 or more 2.00
Once passed the supervisor’s control at bank-wide level, a trading desk can use the
Internal Model Approach (IMA) if it does not experiences either more than 12
exceptions at the 99th percentile or 30 exceptions at the 97.5th percentile.
Backtesting Evaluation for IMA
Alpha 1% Alpha 2.5% Alpha 5%
Expected VaR
Exceed
2.5 6.3 12
Actual VaR Exceed
5 6 6
Unconditional Coverage 95% H_0 = Correct Exceedances
LR.uc Statistic 1.917 0.015 4.477
LR.uc Critical 3.841 3.841 3.841
LR.uc p-value 0.166 0.903 0.034
Reject Null NO NO YES
Conditional Coverage 95% H_0 = Correct and Independent
Exceedances
LR.cc Statistic 2.12 0.309 4.770
LR.cc Critical 5.991 5.991 5.991
LR.cc p-value 0.346 0.857 0.092
Reject Null NO NO NO
The 1%, 2.5% and 5% daily VaR is showed, based on a sample of 252 observations (prior 12 months of the sample), still using the GARCH volatility, to determine
if the bank can use the IMA.
VaR 1 Year Comparison
Alpha 1% Alpha 2.5% Alpha 5%
MA(1)-GARCH(1,1) 1 Year -2.93% -2.28% -1.86%*

15. 1 Year VaR Backtesting

16. VaR drawbacks: why the change?
The decision by the Basel Committee to switch from VaR at 99% to Expected Shortfall at 97.5% is driven by two
main reasons.
• First, it is not a coherent risk measure as it lacks sub-additivity property, that is a property such that the risk of a
portfolio which includes several assets must be less than the sum of the risks of the individual assets. So, VaR
does not reflect the risk reduction from diversification.
• Then, does not capture tail risk, that is the risk that arises when the losses that exceed the VaR are significant.
Value at Risk instead completely relies on the probability of events and does not consider their severity, that is the
height of the incurred loss.

17. Expected Shortfall and backtesting
In order to capture the tail risk and overcome also the lack of
subadditivity of VaR, BCBS provides the Expected Shortfall at
97.5th percentile as risk measure.
Different Expected Shortfall are computed at 99th, 97.5th and
95th percentile, based on GARCH and EWMA VaR, as well as
on the prior year returns,, as imposed by FRTB.
Expected Shortfall Comparison
Alpha 1% Alpha 2.5% Alpha 5%
MA(1)-
GARCH(1,1)
-2.84% -2.61% -2.29%
EWMA with
normal distribution
-2.49% -2.42% -2.23%
MA(1)-
GARCH(1,1) 1 Year
-3.34% -2.89% -2.63%
Alpha 1% Alpha 2.5% Alpha 5%
MA(1)-
GARCH(1,1)
0 0 0
EWMA with
normal distribution
0 0 1.11e-16
MA(1)-
GARCH(1,1) 1 Year
0 0 0
But Expected Shortfall has also some drawbacks, one of the
major ones is its difficult to be back-tested, mainly due to the
fact that ES do not has the elicitability propriety that allows to
compare simulated estimates with observed data.
To overcome this issue, Acerbi and Szekely (2014) proposed
an Expected Shortfall backtesting which does not require the
elicitability propriety. We carry out the backtest at 99%,
97.5% and 95% based on different VaR backtesting.
A value of Z1 equal to 0 indicates that backtest is passed

18. Conclusions
• Regarding VaR and Expected Shortfall, VaR at 97.5th percentile, and not at 99th.,computed assuming a
MA(1)-GARCH(1,1) distribution for returns, turned out to be the most reliable in terms of backtesting
considering the prior year’s returns, while the EWMA VaR estimates seem to be quite weak at
backtesting.
• Concluding, the empirical results obtained, even though based on a single index but still based on the
Basel standards, confirmed the validity of the choice by the BCBS to use the ES 97.5% in order to
determine the capital requirement of the financial institutions.

19. Appendix: S&P500 during the Covid-19 crisis
S&P500 index from 1.07.2019 to 1.07.2020 on a daily base, n. 254 observations.
VaR Comparison
Alpha 1% Alpha 2.5% Alpha 5%
-6.95% -4.51% -3.09%*
Given the VaR excessive values, relying on the fact that Basel Committee imposes the last year’s returns as minimum sample
to compute daily VaR and ES, the author's opinion is that the better solution should be enlarging the sample of observations
over the last year, to better balance the default risk hedging and the capital requirement
Mean Median Min Max
0.000197 0.001236 -127652 0.089683
Std Dev Coeff. of Var. Skewness Kurtosis
0.02127809 107.7685 -0.8734532 12.6606

20. Thank you for the attention